12 HIGHER DERIVATIVES 179

5. (a) Let / be a strictly increasing continuous function in an interval [0, a], such that
/(O) = 0; let g be the inverse mapping, which is continuous and strictly increasing in the

[a /»/(fl)

interval [0,/(a)]. Show that /(/) dt = (a - g(u)) du (apply Problem 4 to the func-
Jo Jo

tion equal to 1 for 0 ^ x ^ a, 0 ^ y </(*), to 0 for 0 ^ x ^ a,f(x) < y ^f(a)).
(b) Show that if 0 ^ x ^ a and 0 =^ y =^/(a), the following inequality holds

the two sides are equal if and only if y =f(x).
(c) Deduce from (b) the inequalities

-1 for *> 0,

for x ^ 0, y^Q, p>!9 q>\9 - + -=1,

p q

a > 0, b > 0 and (pa)q(qb)p ^ 1 .

12. HIGHER DERIVATIVES

Suppose / is a continuously differentiable mapping of an open subset
A of a Banach space E into a Banach space F. Then D/is a continuous
mapping of A into the Banach space ££(E; F). If that mapping is diiferen-
tiable at a point XQ e A (resp. in A), we say that/is twice differentiable at x0
(resp. in A), and the derivative of D/ at XQ is called the second derivative of
/at*0» and written//r(x0) or D2/(x0). This is an element of <£(E; ^(E; F));
but we have seen (5.7.8) that this last space is naturally identified with the
space J&?(E, E;F) (written jSf2(E; F)) of continuous bilinear mappings of
E x E into F: we recall that this is done by identifying u e &(E; &(E; F))
to the bilinear mapping (s,t)'-*(u-s)-t; this last element will also be
written u • (s, t).

(8.12.1) Suppose f is twice differentiable at XQ; then, for any fixed ZeE,
the derivative of the mapping x-+Df(x) - t of A into F, at the point XQ, is

If we observe that x -» Df(x) - 1 is composed of the linear mapping
u -» u - 1 of y (E ; F) into F and of the mapping x -+ D/(JC) of E into JSP (E ; F)
the result follows from (8.2.1) and (8.1.3).